Cumulative Distribution Functions (VCE SSCE Mathematical Methods): Quizzes

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Cumulative distribution functions
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What does the cumulative distribution function F(x)F(x) represent?

Probability that XX is at most xx

For a continuous random variable on [c,d][c, d], how is F(x)F(x) calculated?

F(x)=cxf(t)dtF(x) = \int_c^x f(t) \, dt

What is the value of F(c)F(c) at the lower bound cc of the domain?

F(c)=0F(c) = 0

What is the value of F(d)F(d) at the upper bound dd of the domain?

F(d)=1F(d) = 1

If x1x2x_1 \leq x_2, what can we say about F(x1)F(x_1) and F(x2)F(x_2)?

F(x1)F(x2)F(x_1) \leq F(x_2)

What is the relationship between the CDF F(x)F(x) and PDF f(x)f(x)?

F(x)=f(x)F'(x) = f(x)

What can be said about the continuity of the CDF for a continuous random variable?

CDF is always continuous

What is the typical shape of a cumulative distribution function?

S-shaped curve

For PDF defined on all real numbers, what is F(x)F(x)?

F(x)=xf(t)dtF(x) = \int_{-\infty}^x f(t) \, dt

If f(x)=5x2f(x) = \frac{5}{x^2} for x5x \geq 5, what is F(x)F(x)?

F(x)=15xF(x) = 1 - \frac{5}{x}

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